Turunan Hasil Bagi Fungsi-fungsi

Jika f(x) = u(x)/v(x), dengan v(x) ≠ 0; u(x) dan v(x) adalah masing-masing fungsi dari x yang memiliki turunan u’(x) dan v’(x), maka turunan fungsi f(x) adalah

$f'(x)= \frac{u'(x)\times v(x)-u(x)\times v'(x) }{v^2(x)} $

Bukti:

$f'(x) = \lim\limits_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}$

$f'(x) = \lim\limits_{h\rightarrow 0}\left [\frac{\frac{u(x+h)}{v(x+h)}-\frac{u(x)}{v(x)}}{h}  \right ]$

$f'(x) = \lim\limits_{h\rightarrow 0} \left [ \frac{u(x+h)v(x)-u(x)v(x+h)}{hv(x+h)v(x)} \right ]$

$f'(x) = \lim\limits_{h\rightarrow 0} \left [ \frac{u(x+h)v(x)-u(x)v(x)-u(x)v(x+h)+u(x)v(x)}{hv(x+h)v(x)} \right ]$ 

$f'(x) = \lim\limits_{h\rightarrow 0} \left [ \frac{\frac{u(x+h)-u(x)}{h}v(x)-\frac{v(x+h)-v(x)}{h}u(x)}{h} \right ]$

$f'(x) =  \frac{\left (\lim\limits_{h\rightarrow 0}\frac{u(x+h) -u(x)}{h}  \right )  v(x)  \ - \left (\lim\limits_{h\rightarrow 0}\frac{v(x+h)-v(x)}{h}  \right )u(x)}{\lim\limits_{h\rightarrow 0}v(x+h)v(x)} $

$f'(x) = \frac{u'(x)v(x)-u(x)v'(x)}{v(x)v(x)}$

$f'(x) = \frac{u'(x)v(x)-u(x)v'(x)}{v^2(x)}$. Terbukti

Contoh Soal 1

Carilah turunan f'(x) dari setiap fungsi berikut ini.

a. $f(x)=\frac{6}{x^3-3x}$

b. $f(x)=\frac{x^2+x-1}{3x^2+x+5}$

c. $f(x)=\frac{2x}{2-\sqrt{x}}$

Jawab:

a. $f(x)=\frac{6}{x^3-3x}$

u = 6 → u' = 0 

$v= x^3 -3x$ → $v'=3x^2 - 3$, maka

$f'(x) = \frac{u'(x)v(x)-u(x)v'(x)}{v^2(x)}$

$f'(x) = \frac{(0)(x^3 - 3x)-(6)(3x^2 - 3)}{(x^3 - 3x)^2}$

$f'(x) = \frac{-18(x^2 - 1)}{(x^3 - 3x)^2}$

b. $f(x)=\frac{x^2+x-1}{3x^2+x+5}$

$u= x^2 +x-1$ → u' = 2x + 1 

$v= 3x^2 +x+5$ → v' = 6x + 1, maka

$f'(x) = \frac{(2x+1)(6x+1)-(x^2 + x - 1)(6x + 1)}{(3x^2 + x + 5)^2}$

$f'(x) = \frac{6x^3+2x^2 + 10x +x + 5-6x^3-x^2-6x^2-x+6x+1}{(3x^2 + x + 5)^2}$

$f'(x) = \frac{-2x^2 + 16x+6)}{(3x^2 + x + 5)^2}$

c. $f(x)=\frac{2x}{2-\sqrt{x}}$

u = 2x → u' = 2  

$v=2-\sqrt{x}=2-x^{\frac{1}{2}}$ → $v'=-\frac{1}{2}x^{-\frac{1}{2}}$, maka

$f'(x) = \frac{(2)(2-\sqrt{x})-(2x)(-\frac{1}{2}x^{-\frac{1}{2}})}{(2-\sqrt{x})^2}$

$f'(x) = \frac{4-2\sqrt{x}+\sqrt{x}}{(2-\sqrt{x})^2}$

$f'(x) = \frac{4-\sqrt{x}}{(2-\sqrt{x})^2}$